A unique sorting algorithm with linear time & space complexity

IJRET: International Journal of Research in Engineering and Technology eISSN:2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 264
A UNIQUE SORTING ALGORITHM WITH LINEAR TIME & SPACE
COMPLEXITY
Sanjib Palui1
, Somsubhra Gupta2
1
PLP, CVPR Unit, Indian Statistical Institute, WB, India
2
HOD, Information Technology, JISCE, WB, India
Abstract
Sorting a list means selection of the particular permutation of the members of that list in which the final permutation contains
members in increasing or in decreasing order. Sorted list is prerequisite of some optimized operations such as searching an
element from a list, locating or removing an element to/ from a list and merging two sorted list in a database etc. As volume of
information is growing up day by day in the world around us and these data are unavoidable to manage for real life situations,
the efficient and cost effective sorting algorithms are required. There are several numbers of fundamental and problem oriented
sorting algorithms but still now sorting a problem has attracted a great deal of research, perhaps due to the complexity of solving
it efficiently and effectively despite of its simple and familiar statements.
Algorithms having same efficiency to do a same work using different mechanisms must differ their required time and space. For
that reason an algorithm is chosen according to one’s need with respect to space complexity and time complexity. Now a day,
space (Memory) is available in market comparatively in cheap cost. So, time complexity is a major issue for an algorithm. Here,
the presented approach is to sort a list with linear time and space complexity using divide and conquer rule by partitioning a
problem into n (input size) number of sub problems then these sub problems are solved recursively. Required time and space for
the algorithm is optimized through reducing the height of the recursive tree and reduced height is too small (as compared to the
problem size) to evaluate. So, asymptotic efficiency of this algorithm is very high with respect to time and space.
Keywords: sorting, searching, permutation, divide and conquer algorithm, asymptotic efficiency, space complexity,
time complexity, and recursion.
--------------------------------------------------------------------***----------------------------------------------------------------------
1. INTRODUCTION
An algorithm [1] [2] [3] [6] [7] is a finite set of instructions,
that if followed, accomplish a particular task. Algorithm
must satisfy some characteristic like having input, output,
definiteness, finiteness, effectiveness.
Sorting [2] [5] [8] a list means a particular permutation of a
given sequence in which the elements of the sequence are
arranged in increasing/decreasing order. Sorting algorithms
used in computer science are often classified by:
 Computational complexity [5] [9] [10] (worst, average
and best behavior) of element comparisons in terms of
the size of the list.
 Computational complexity of swaps (for "in place"
algorithms) are sometimes characterized in terms of
the performances that the algorithms yield and the
amount of time that the algorithms take.
 Requirements of memory [11] [12] and other computer
resources.
 Recursion [13]. Some algorithms are either recursive
or non-recursive, while others may be both.
 Stability: stable sorting algorithms maintain the
relative order of records with equal keys i.e. values.
 Whether or not they are a comparison sort. A
comparison sort examines the data only by comparing
two elements with a comparison operator.
 General method: insertion, exchange, selection,
merging, etc.
 Adaptability: Whether or not the pre-sortedness of the
input affects the running time. Algorithms that take
this into account are known to be adaptive.
Now a day there are many effective sorting algorithms.
Although lots of researchers are working on this, but unlike
other field of research, number of proposed new, innovative
and cost effective work is very few in the field of sorting
algorithm.
We have designed and applied one sorting algorithm to
achieve linear time complexity. In this paper, this new
algorithm is proposed. As compared to existing algorithm, it
gives better result and also it has linear time and space
complexity, we named this work of algorithm as --“A
Unique Sorting Algorithm with Linear Time & Space
Complexity”.
Here, our work is organized as follows: PREVIOUS
RELATED WORKS is given in section 2, ALGORITHM
OF PROPOSED WORK is in section 3, ANALYSIS OF
THE ALGORITHM is in section 4, and finally conclusion is
given on section 5 and then REFERENCEs.

_______________________________________________________________________________________
2. PREVIOUS RELATED WORKS
Some well known sorting algorithm are bubble sort,
selection sort, insertion sort ,merge sort, quick sort ,heap
sort ,radix sort , cocktail sort ,shell sort etc. All these
algorithms can be classified according to their average case
and worst case time complexity (asymptotic complexity-big
theta notation-θ and big oh notation – O respectively). Time
complexity of some algorithms are given in Table 1 along
with the stability of these algorithms i.e. they are stable or
not. Here (with respect to the Table 1) n is the input size of
the list to be sorted, d is number of digit in the largest
number among inputs and k is all possible digit/word (for
example- k=10 as decimal).
Some comparative study [11] [14] [15] [16] have been
carried out in this field and situations of better suitemate for
these algorithms (Table 1) are clearly notified. Depend on
applications or data structure, which among sorting
algorithm or modified version of an existing algorithm has
less complexity than the original one or which among these
algorithms is the best fitted in a particular circumstances are
drawn on some proposed approach [17] [18] [19] [20].
Table -1: Information about Some algorithms
Name of the
Algorithm
Time Complexity Stable?
Average case Worst Case
Bubble sort Θ(n2
) O(n2
) Yes
Selection sort Θ(n2
) O(n2
) No
Insertion sort Θ(n2
) O(n2
) Yes
Merge sort Θ(n log2 n) O(n log2 n) Yes
Quick sort Θ(n log2 n) O(n2
) No
Bucket sort Θ(d(n+k)) O(n2
) Yes
Heap sort Θ(n log2 n) O(n log2 n) No
Depending on inputs and some predefined conditions, some
new algorithms [18] [21] [22] have been projected to
achieve better complexity. Some algorithm is designed for
linear complexity [23] [24] but they can be executed in the
system platform having some special characteristics as these
algorithms demand for that. Poll sort, simple pancake sort,
bead sort are the example of this type of sorting algorithms.
Pancake sorting algorithm is not stable but has linear time
complexity. Bead sort algorithm requires special hardware
design to execute. Poll sort algorithm takes linear-time to
execute but it is an analog algorithm for sorting a sequence
of items, requiring O(n) stack space, and the sorting
algorithm is stable but n (number of elements in input list)
parallel processors are required.
There is an algorithm called Randomized Select [24] [25]
algorithm for sorting. The expected running time of this
algorithm is θ (n); a linear asymptotic running time where n
is the input size of the problem. This algorithm works like
Randomized Quick-sort [26] (where pivot element is select
randomly from the list).Two main constraint of this
algorithm
 All the elements in the input sub-problem are
distinct.
 Partition is done based on random selection of an
element.
Complexities of algorithms are strictly dependent on input
size, caches performance etc., briefly description about
complexities of algorithms are given some of the noble work
[28] [29] [30].
3. ALGORITHM OF PROPOSED WORK
The pseudo-code of our work, --“A Unique Sorting
Algorithm with Linear Time & Space Complexity” is given
below
Sort (A, lb, ub)
Here A is a 1-D input list of decimal integer with n elements
in memory where n=ub-lb+1 and ub=upper bound, lb=lower
bound of the list.
Begin
1. if (lb<ub) then,
2. find min and max
3. if (min!=max) then,
4. set n = ub-lb+1
5. create n number of empty List
6. set div = (max - min) /n +1
7. for i=lb to ub by 1do,
8. j= (A[i] - min)/div
9. add A[i] to the jth
List
10. end for
11. set k = lb
12. for i =0 to n by 1 do,
13. set l = k
14. set size =number of elements in ith
List
15. for j =0 to size by 1 do,
16. A [k]= jth
element of ith
List
17. set k = k +1
18. end for
19. If ((l <k-1) && (! ((l = = lb)&&(k = = ub))))
20. call sort( A, l, k -1)
21. end if
22. end for
23. end if
24. end if
End
Algorithm1.
min and max in line number 2 in Algorithm1 are minimum
and maximum number from the list respectively. Here we
have designed one variable for every “sort (x, y, z)” function
call, called div which plays major role in the above
mentioned pseudo code of the designed sorting algorithm.
Every element of the input list forms its own index as it will
be in the output list with the help of div variable[line
number 8 of Algorithm1].Then all the elements of the input
list with same indexes (as it is generated in line number 8 of
Algorithm1) are solved as sub problems[27]. Elements with
same index are treated here as one sub problem. For
increasing order output----

_______________________________________________________________________________________
 Every sub problem (except left most) as single unit
has all smaller elements from the input list in its left.
 Every sub problem (except right most) as single unit
has all larger elements from the input list in its right.
 Every sub problem can have 0 to n-1 elements.
 If number of elements in one sub problem is one it
takes unit time cost to sort.
 Ideally list with n elements are partitioned in to n sub
problems each of which have one element.
 It takes linear time complexity, O (n).
 Relative positions of two elements having same value
are not changed, so this algorithm is stable.
All other strategies that are followed to execute this
algorithm properly and efficiently-
 If the input list is combination of positive and
negative integers, then two list are formed i.e. one list
for negative numbers and another list for positive
numbers. Then all the elements of the list of negative
numbers are made positive. After performing
algorithmic operations to make them as sorted, sign
of elements are changed again and the list is reversed.
Then operations are performed on positive list. At
last, these two lists are concatenated.
 If the numbers are real, this algorithm can be applied
easily. Problem is divided into sub problems
according to their absolute value and if absolute
values are same for all elements then partition is done
based on their precision value.
4. ANALYSIS OF THE ALGORITHM
From the number statistics if the numbers are in uniform
distribution then almost no recursions are happened,
otherwise after first partition of the array, it will make
greater than or equal to two uniform distributed array where
complexity is linearly dependant on n. Here, after partition
of the array some input elements are taken place in the array
like in quick sort one element is fixed in exact position.
Here number of fixed elements is 1 to n where n is the
number of elements to be sorted.
4.1 Time Complexity
Asymptotic time complexities [3] [4] of this algorithm in
different cases are described here-
4.1.1 Best Case
If the elements are uniform distributed, then ideally
problems are divided in to n sub problems. So no recursive
call is executed because almost each sub problem has one
element.
So, the required time is
T (n) =c +T(1)+T(1)+T(1)+T(1)…………n times
≡c + O (n) where c=constant time
≡ O (n) as n is very large
4.1.2 Average Case
If number of elements n becomes very high, in first function
call, divided sub-problems will be almost uniformly
distributed. Let, here recursive call is happened m times
where (m<n) and sub-problems have n1, n2… nm elements
respectively.
So, n1+n2+…..+nm +p= n ---- Eq. (1)
where p is the number of elements that are already taken
place in the input list as sorted elements. So, time
complexity is
T(n)=c.n+ T(n1)+T(n2)+……+T(nm)
Where
c=constant time
=c. n+ c.n1 +c.n2+ ……+c.nm
[From the best case as the sub problems are uniform
distributed of , ,…, elements, there is rare chance for
every sub problem to give average case time complexity ]
= c. n+ c. ( + +……+ )
=c. n+ c. n [from, Eq. (1) ]
=2cn
≡O(n)
4.1.3 Worst Case
The algorithm gives worst case time complexity when,
 Inputs elements are randomly distributed.
 All elements except the largest one of the input list
have values less than the value of div as it is
generated in line number 6 in our algorithm1 and it is
happen again and again for every sub problem.
 The input series is one of all possible permutation of
the elements of the series-----
a, i*(i-1)th
term +c ∀ i=2 to n - --Eq.(2)
where a=starting element of this series and having a
positive value and c > = 0.
4.1.3.1 Case-I
According to the proposed algorithm, at that situation, time
complexity is
T(n)=c .n+ T(n-1)
≡O(n2
)
which is possible theoretically but not in real life because
we are talking about asymptotic time complexity. So,
number of input elements is very very high. Every real life
database collects and stores similar kinds of data and it is a
very rare chance that at least some of elements in the sorted
output series satisfy Eq. (2).
So, for every real life problem this algorithm gives average
case time complexity.

_______________________________________________________________________________________
4.1.3.2 Case-II
Depend on that situation of the worst case we have added
some more instruction with the original above mentioned
algorithm (i.e. Algorithm1) in between line number 3 and
line number 4.
 Second largest number is found out along with the
largest number.
 Checking is done according to the series of Eq. (2).
If match then max (variable of the algorithm1) is
set to second largest number instead of the largest
number.
After this, it is seen that the problem is partitioned at least in
to three sub problems. It is empirically designed in such a
way that time complexity of the worst case for this
algorithm become .
4.2 Space Complexity
Space complexities [4] [8] in different cases are fully
dependent on number of times the subroutine (as described
in Algorithm1) is called recursively and input size for that
sub-problem for which this subroutine is called.
When input size is n ,this algorithm needs more memory for
the n numbers of list which are initially empty and store the
elements of same index in the time of inputs being
processed (from Algorithm1-line number 5 and line
number 9). Now total number of elements in the n numbers
of list is n.
So, required memory unit for every subroutine call is
n (for input elements) +n (for index lists)+c
= 2*n +c
Where c = memory unit for other variables
So, asymptotic space complexity is also linearly dependent
on input size as this algorithm is implemented to reduce the
depth of recursion tree.
Complexity and stability details of presented algorithm are
in Table-2---
Table -2: Information about Presented algorithm
Time Complexity Space
Complexity
Stable?
Best Case O(n)
Same as
time
complexity
Yes
Average
Case
O(n)
Worst
Case
 O(n2
) or
Only for Eq.(2)
 O(n)
Some experimental results with respect to input sizes and
required times of this algorithm are given in Table-3 as
compared to the results of some other well known
algorithms where computer environment, paradigm, inputs
are same.
Table -3: Experiment Results
Input
Size
Required Time in Millisecond
Bubbl
e Sort
Selectio
n Sort
Insertio
n Sort
Merg
e Sort
Proposed
Approac
h
15000 1248 442 328 7 7
30000 4863 1746 1325 22 13
60000 19904 7056 6883 71 25
90000 43550 15750 14921 150 27
12000
0
76099 28917 25132 265 36
15000
0
12241
6
44925 38081 394 45
5. CONCLUSIONS
This presented algorithm is implemented successfully
through a repeatable task of design, carrying out and
analyze. The advantages of this algorithm are its speed,
requirement of less memory than existing one and stability.
Selection of sorting algorithm is application and situation
dependent but this algorithm works well in every field. For
real life sorting problem, time complexity & space
complexity of this algorithm are linear.
There is a broad future scope to experiment the proposed
algorithm for finding out short-coming (if any) based on
some uncovered real life test suits with solutions.
REFERENCES
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Computer Programming: Vol.1, Addison-Wesley, 1968.
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Edition). Sams, 978-0-672-32453-6..

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BIOGRAPHIES
Sanjib Palui has completed his bachelor
degree in Information Technology and
master degree in Software Engineering
in the year of 2012 and 2014
respectively. Then he has joined Indian
Statistical Institute as Project Linked
Person in the unit of Computer Vision & Pattern
Recognition (CVPR). He is working on Data Structure &
algorithm, Image Processing and Optical Character
Recognition fields.
Dr. Somsubhra Gupta is presently the
Assistant Professor of the Department of
Information Technology, JIS College of
Engineering (An Autonomous
Institution). He is graduated from the
University of Calcutta and completed his Masters from
Indian Institute of Technology, Kharagpur. He received Ph.
D. award entitled “Multiobjective Decision Making in
Inexact Environment using Genetic Algorithm: Theory and
Applications” form University of Kalyani. His area of
teaching is Algorithm and allied domains and research area
is Machine Intelligence. In research, he has around 56
papers including Book Chapters so far in National /
International Journal / Proceedings and over 40 citations. He
is Principal Investigator / Project Coordinator to some
Research projects (viz. RPS scheme AICTE). He was the
Convener of International Conference on Computation and
Communication Advancement (IC3A-2013). He is invited in
the Technical Programme Committee of number of
Conferences, delivered as an invited speaker, an INS
(Elsevier Science) reviewer and attended NAFSA-2013
conference of international educators at St. Louis, Missouri,
USA

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